Preparation

Cook shallots and ginger in oil in a wide 5-quart heavy pot over medium heat,
stirring frequently, until softened, about
5 minutes. Add tomatoes and red-pepper flakes and cook, stirring, 1 minute.
Pour in carrot juice and boil, uncovered, until liquid is reduced by about half, about 10 minutes.

Add mussels and cook, covered, stirring occasionally, until mussels just open wide, 4 to 6 minutes, transferring as cooked to a bowl. (Discard any mussels that remain unopened after 6 minutes.) Stir in lime juice and sprinkle with cilantro.